An Interview with F. William Lawvere

You have written a paper, published for the first time in 1986, entitled “Takingcategories seriously”1. Why should we take categories seriously ?

In all those areas where category theory is actively used the categorical conceptof adjoint functor has come to play a key role. Such a universal instrument forguiding the learning, development, and use of advanced mathematics does notfail to have its indications also in areas of school and college mathematics, in themost basic relationships of space and quantity and the calculations based on thoserelationships. By saying “take categories seriously”, I meant that one should seek,cultivate, and teach helpful examples of an elementary nature.

The relation between teaching andresearch is partly embodied in simplegeneral concepts that can guide theelaboration of examples in both. No-tions and constructions, such as thespectral analysis of dynamical systems,have important aspects that can beunderstood and pursued without thecomplications of limiting the modelsto specific classical categories. Theapplication of some simple generalconcepts from category theory canlead from a clarification of basic con-structions on dynamical systems to aconstruction of the real number systemwith its structure as a closed cate-gory; applied to that particular closed

F. William Lawvere (Braga, March 2007)

category, the general enriched category theory leads inexorably to embedding the-orems and to notions of Cauchy completeness, rotation, convex hull, radius, and

1Revista Colombiana de Matematicas 20 (1986) 147-178. Reprinted in Repr. Theory Appl. Categ. 8(2005) 1-24 (electronic).

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geodesic distance for arbitrary metric spaces. In fact, the latter notions presentthemselves in such a form that the calculations in elementary analysis and geome-try can be explicitly guided by the experience that is concentrated in adjointness.It seems certain that this approach, combined with a sober application of the his-torical origin of all notions, will apply to many more examples, thus unifying ourefforts in the teaching, research, and application of mathematics.

I also believe that we should take seriously the historical precursors of categorytheory, such as Grassman, whose works contain much clarity, contrary to his rep-utation for obscurity.

Other than Grassman, and Emmy Noether and Heinz Hopf, whom Mac Lane usedto mention often, could you name other historical precursors of category theory ?

The axiomatic method involves concentrating key features of ongoing applications.For example, Cantor concentrated the concept of isomorphism, which he had ex-tracted from the work of Jakob Steiner on algebraic geometry. The connection ofCantor with Steiner is not mentioned in most books; there is an unfortunate ten-dency for standard works on the history of science to perpetuate standard myths,rather than to discover and clarify conceptual analyses. The indispensable “uni-verse of discourse” principle was refined into the idea of structure carried by anabstract set, thus making long chains of reasoning more reliable by approachingthe ideal that “there is nothing in the conclusion that is not in the premise”. Thatvision was developed by Dedekind, Hausdorff, Frechet, and others into the 20thcentury mathematics.

Besides the portraits of the inventors of category theory, Eilenberg and Mac Lane,the front cover of our book “Sets for Mathematics”, written in collaboration withRobert Rosebrugh, contains the portraits of Cantor and Dedekind.

The core of mathematical theories is in the variation of quantity in space and inthe emergence of quality within that. The fundamental branches such as differen-tial geometry and geometric measure theory gave rise to the two great auxiliarydisciplines of algebraic topology and functional analysis. A great impetus to theircrystallization was the electromagnetic theory of Maxwell-Hertz-Heaviside and thematerials science of Maxwell-Boltzmann. Both of these disciplines and both ofthese applications were early made explicit in the work of Volterra. As pointed outby de Rham to Narasimhan, it was Volterra who in the 1880’s not only proved thatthe exterior derivative operator satisfies d2 = 0, but proved also the local existencetheorem which is usually inexactly referred to as the Poincare lemma; these resultsremain the core of algebraic topology as expressed in de Rham’s theorem and inthe cohomology of sheaves.

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Commonly, the codomain category for a quantitative functor on X is a categoryMod(X) of linear structures in X itself; thus it is most basically the nature ofthe categories X of spaces that such systems of quantities have as domain whichneeds to be clarified. Concentrating the contributions of Volterra, Hadamard, Fox,Hurewicz and other pioneers, we arrive at the important general idea that suchcategories should be Cartesian closed. For example, the power-set axiom for sets isone manifestation of this idea – note that it is not “justified” by the 20th centuryset-theoretic paraphernalia of ordinal iteration, formulas, etc., since it, togetherwith the axiom of infinity, must be in addition assumed outright. Hurewicz was,like Eilenberg, a Polish topologist, and his work on homotopy groups, presentedin a Moscow conference, was also pioneer; too little known is his 1949 lecture onk-spaces, the first major effort, still used by algebraic topologists and analysts,to replace the “default” category of topological spaces by a more useful Cartesianclosed one.

Speaking of Volterra, it reminds us that you have praised somewhere2 the work ofthe Portuguese mathematician J. Sebastiao e Silva. Could you tell us somethingabout it ?

Silva was one of the first to recognize the importance of bornological spaces as aframework for functional analysis. He thus anticipated the work of Waelbroeckon smooth functional analysis and prepared the way for the work of Douady andHouzel on Grauert’s finiteness theorem for proper maps of analytic spaces. More-over, in spite of my scant Portuguese, I discern in Silva a dedication to the closerelation between research and teaching in a spirit that I share.

Where did category theory originate ?

The need for unification and simplification to render coherent some of the manymathematical advances of the 1930’s led Eilenberg and Mac Lane to devise thetheory of categories, functors and natural transformations in the early 1940’s. Thetheory of categories originated in their GTNE article3, with the need to guide com-plicated calculations involving passage to the limit in the study of the qualitativeleap from spaces to homotopical/homological objects. Since then it is still activelyused for those problems but also in algebraic geometry, logic and set theory, modeltheory, functional analysis, continuum physics, combinatorics, etc.

2F. W. Lawvere, Volterras functionals and covariant cohesion of space, Suppl. Rend. Circ. Mat.Palermo, serie II, 64 (2000) 201-214.

3S. Eilenberg and S. Mac Lane, General Theory of Natural Equivalences, Trans. Amer. Math. Soc. 58(1945) 231-294.

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G. M. Kelly, S. Mac Lane and F. W. Lawvere(CT99 conference, held in Coimbra on the occasion of the 90th birthday of

Saunders Mac Lane; photo courtesy of J. Koslowski)

Mac Lane entered algebraic topology through his friend Samuel Eilenberg. To-gether they constructed the famous Eilenberg-Mac Lane spaces, which “representcohomology”. That seemingly technical result of geometry and algebra required,in fact, several striking methodological advances: (a) cohomology is a “functor”,a specific kind of dependence on change of domain space; (b) the category wherethese functors are defined has as maps not the ordinary continuous ones, but ratherequivalence classes of such maps, where arbitrary continuous deformations of mapsserve to establish the equivalences; and (c) although in any category any fixed ob-ject K determines a special “representable” functor that assigns, to any X, theset [X, K] of maps from X to K, most functors are not of that form and thus it isremarkable that the particular cohomological functors of interest turned out to beisomorphic to H∗(X) = [X, K] but only for the Hurewicz category (b) and only forthe spaces K of the kind constructed for H∗ by Eilenberg and Mac Lane. All thoseadvances depended on the concepts of category and functor, invented likewise in1942 by the collaborators! Even as the notion of category itself was being madeexplicit, this result made apparent that “concrete” categories, in which maps aredetermined by their values on points, do not suffice.

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Already in GTNE it was pointed out that a preordered set is just a category withat most one morphism between any given pair of objects, and that functors be-tween two such categories are just order-preserving maps; at the opposite extreme,a monoid is just a category with exactly one object, and functors between two suchcategories are just homomorphisms of monoids. But category theory does not restcontent with mere classification in the spirit of Wolffian metaphysics (although afew of its practitioners may do so); rather it is the mutability of mathematicallyprecise structures (by morphisms) which is the essential content of category theory.If the structures are themselves categories, this mutability is expressed by func-tors, while if the structures are functors, the mutability is expressed by naturaltransformations.

The New York Times, in its 1998 obituary of Eilenberg, omitted completely Eilen-berg’s role in the development of category theory.

Yes, and the injustice was only slightly less on the later occasion of Mac Lane’sobituary, when the Times gave only a vague account.

P. T. Johnstone, F. W. Lawvere and P. Freyd

(CT06 conference, White Point; photo courtesy of J. Koslowski)

In a letter to the NYT in February 1998, written jointly with Peter Freyd, you com-plained about that notable omission. In it you stress that the Eilenberg-Mac Lane

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“discovery in 1945 of the theory of transformations between mathematical categoriesprovided the tools without which Sammy’s important collaborations with Steenrodand Cartan would not have been possible. That joint work laid also the basis forSammy’s pioneering work in theoretical computer science and for a great manycontinuing developments in geometry, algebra, and the foundations of mathemat-ics. In particular, the Eilenberg-Mac Lane theory of categories was indispensableto the 1960 development, by the French mathematician Alexander Grothendieck, ofthe powerful form of algebraic geometry which was an ingredient in several recentadvances in number theory, including Wiles’ work on the Fermat theorem”. Couldyou give us a broad justification of why category theory may be so useful ?

Everyday human activities such as building a house on a hill by a stream, layinga network of telephone conduits, navigating the solar system, require plans thatcan work. Planning any such undertaking requires the development of thinkingabout space. Each development involves many steps of thought and many relatedgeometrical constructions on spaces. Because of the necessary multistep natureof thinking about space, uniquely mathematical measures must be taken to makeit reliable. Only explicit principles of thinking (logic) and explicit principles ofspace (geometry) can guarantee reliability. The great advance made by the theoryinvented 60 years ago by Eilenberg and Mac Lane permitted making the principlesof logic and geometry explicit; this was accomplished by discovering the commonform of logic and geometry so that the principles of the relation between the two arealso explicit. They solved a problem opened 2300 years earlier by Aristotle with hisinitial inroads into making explicit the Categories of Concepts. In the 21st century,their solution is applicable not only to plane geometry and to medieval syllogisms,but also to infinite-dimensional spaces of transformations, to “spaces” of data, andto other conceptual tools that are applied thousands of times a day. The form ofthe principles of both logic and geometry was discovered by categorists to rest on“naturality” of the transformations between spaces and the transformations withinthought.

What are your recollections of Grothendieck ? When did you first meet him ?

I had my first encounter with him at the ICM (Nice, 1970) where we were bothinvited lecturers. I publicly disagreed with some points he made in a separatelecture on his “Survival” movement, so that he later referred to me (affectionately,I hope) as the “main contradictor”. In 1973 we were both briefly visiting Buffalo,where I vividly remember his tutoring me on basic insights of algebraic geometry,such as “points have automorphisms”. In 1981 I visited him in his stone hut, in themiddle of a lavender field in the south of France, in order to ask his opinion of a

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project to derive the Grauert theorem from the Cartan-Serre theorem, by provingthe latter for a compact analytic space in a general topos, then specializing to thetopos of sheaves on a parameter space. Some needed ingredients were known, forexample that a compact space in the internal sense would correspond to a propermap to the parameter space externally. But the proof of these results classicallydepends on functional analysis, so that the theory of bornological spaces wouldhave to be done internally in order to succeed. He recognized right away that sucha development would depend on the use of the subobject classifier which, as hesaid, is one of the few ingredients of topos theory that he had not foreseen. Later inhis work on homotopy he kindly referred to that object as the “Lawvere element”.My last meeting with him was at the same place in 1989 (Aurelio Carboni droveme there from Milano): he was clearly glad to see me but would not speak, theresult of a religious vow; he wrote on paper that he was also forbidden to discussmathematics, though quickly his mathematical soul triumphed, leaving me withsome precious mathematical notes.

F. W. Lawvere, A. Heller, R. Lavendhomme (in the back) and A. Carboni (CT99, Coimbra)

But the drastic reduction of scientific work by such a great mathematician, due tothe encounter with a powerful designer religion, is cause for renewed vigilance.

You were born in Indiana. Did you grow up there ?

Yes. I have been sometimes called “the farmboy from Indiana”.

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Did your parents have any mathematical interest ?

No. My father was a farmer.

You obtained your BA degree from Indiana University in 1960. Please tell us alittle bit about your education there. How did you learn about categories ? Weknow that you started out as a student of Clifford Truesdell, a well-known experton classical mechanics.4

I had been a student at Indiana University from 1955 to January 1960. I likedexperimental physics but did not appreciate the imprecise reasoning in some the-oretical courses. So I decided to study mathematics first. Truesdell was at theMathematics Department but he had a great knowledge in Engineering Physics.He took charge of my education there.

Eilenberg had briefly been at Indiana, but had left in 1947 when I was just 10years old. Thus it was not from Eilenberg that I learned first categories, norwas it from Truesdell who had taken up his position in Indiana in 1950 and whoin 1955 (and subsequently) had advised me on pursuing the study of continuummechanics and kinetic theory. It was a fellow student at Indiana who pointed outto me the importance of the galactic method mentioned in J. L. Kelley’s topologybook; it seemed too abstract at first, but I learned that “galactic” referred tothe use of categories and functors and we discussed their potential for unifyingand clarifying mathematics of all sorts. In Summer 1958 I studied TopologicalDynamics with George Whaples, with the agenda of understanding as much aspossible in categorical terms. When Truesdell asked me to lecture for severalweeks in his 1958-1959 Functional Analysis course, it quickly became apparentthat very effective explanations of such topics as Rings of Continuous Functionsand the Fourier transform in Abstract Harmonic Analysis could be achieved bymaking explicit their functoriality and naturality in a precise Eilenberg-Mac Lanesense. While continuing to study statistical mechanics and kinetic theory, at somepoint I discovered Godement’s book on sheaf theory in the library and studied itextensively. Throughout 1959 I was developing categorical thinking on my ownand I formulated research programs on “improvement” (which I later learned hadbeen worked out much more fully by Kan under the name of adjoint functors)and on “galactic clusters” (which I later learned had been worked out and appliedby Grothendieck under the name of fibered categories). Categories would clearlybe important for simplifying the foundations of continuum physics. I concludedthat I would make category theory a central line of my study. The literature often

4C. Truesdell was the founder of the journals Archive for Rational Mechanics and Analysis and Archivefor the History of Exact Sciences.

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mentioned some mysterious difficulty in basing category theory on the traditionalset theory: having had a course on Kleene’s book (also with Whaples) and havingenjoyed many discussions with Max Zorn, whose office was adjacent to mine, Ihad some initial understanding of mathematical logic, and concluded that thesolution to the foundational problem would be to develop an axiomatic theory ofthe Category of Categories.

Why did you choose Columbia University to pursue your graduate studies ?

The decision to change graduate school (even before I was officially a graduatestudent) required some investigation. Who were the experts on category theoryand where were they giving courses on it ? I noted that Samuel Eilenberg appearedvery frequently in the relevant literature, both as author and as co-author with MacLane, Steenrod, Cartan, Zilber. Therefore Columbia University was the logicaldestination. Consulting Clifford Truesdell about the proposed move, I was pleasedto learn that he was a personal friend of Samuel Eilenberg; recognizing my resolvehe personally contacted Sammy to facilitate my entrance into Columbia, and I sentdocuments briefly outlining my research programs to Eilenberg.

The NSF graduate fellowship which had supported my last period at Indianaturned out to be portable to Columbia. The Mathematics Department at Columbiahad an arrangement whereby NSF fellows would also serve as teaching assistants.Thus I became a teaching assistant for Hyman Bass’ course on calculus, i.e. linearalgebra, until January 1961.

When I arrived in New York in February 1960, my first act was to go to the Frenchbookstore and buy my own copy of Godement. In my first meeting with Eilenberg,I outlined my idea about the category of categories. Even though I only took onecourse, Homological Algebra, with Eilenberg, and although Eilenberg was veryoccupied that year with his duties as departmental chairman, I was able to learn agreat deal about categories from Dold, Freyd, Mitchell, Gray; with Eilenberg I hadonly one serious mathematical discussion. Perhaps he had not had time to readmy documents; at any rate it was a fellow student, Saul Lubkin, who after I hadbeen at Columbia for several months remarked that what I had written about hadalready been worked out in detail under the name of adjoint functors, and uponasking Eilenberg about that, he gave me a copy of Kan’s paper.

In 1960 Eilenberg had managed to attract at least ten of the later major contribu-tors to category theory to Columbia as students or instructors. These courses anddiscussions naturally helped to make more precise my conception of the categoryof categories, as did my later study of mathematical logic at Berkeley; however

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the necessity for axiomatizing the category of categories was already evident to mewhile studying Godement in Indiana.

A few months later when Mac Lane was visiting New York City, Sammy introducedme to Saunders, jokingly describing my program as the mystifying “Sets withoutelements”.

In his autobiography5, Mac Lane writes that “One day, Sammy told me he had ayoung student who claimed that he could do set theory without elements. It washard to understand the idea, and he wondered if I could talk with the student. (...) Ilistened hard, for over an hour. At the end, I said sadly, ‘Bill, this just won’t work.You can’t do sets without elements, sorry,’ and reported this result to Eilenberg.Lawvere’s graduate fellowship at Columbia was not renewed, and he and his wifeleft for California.” ...

... I never proposed “Sets without elements” but the slogan caused many misun-derstandings during the next 40 years because, for some reason, Saunders likedto repeat it. Of course, what my program discarded was instead the idea of ele-menthood as a primitive, the mathematically relevant ideas of both membershipand inclusion being special cases of unique divisibility with respect to categoricalcomposition. I argue that set theory should not be based on membership, as inZermelo-Frankel set theory, but rather on isomorphism-invariant structure.

F. W. Lawvere and S. Mac Lane (CT97, Vancouver)

5Saunders Mac Lane, A Mathematical Autobiography, A K Peters, 2005.

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About Mac Lane’s autobiography, note that when Mac Lane wrote it he was al-ready at an advanced age, and according to his wife and daughter, he had alreadyhad several strokes. Unfortunately, the publisher rushed into print on the occa-sion of his death without letting his wife and his daughter correct it, as they hadbeen promised. As a consequence, many small details are mistaken, for exam-ple the family name of Mac Lane’s only grandson William, and Coimbra becameColumbia6, etc. Of course, nobody’s memory is so good that he can rememberanother’s history precisely, thus the main points concerning my contributions andmy history often contain speculations that should have been checked by the editorsand publisher.

With respect to that episode, it is treated briefly in the book, but in a rathercompressed fashion, leading to some inaccuracies. The preliminary acceptance ofmy thesis by Eilenberg was encouraged by Mac Lane who acted as outside readerand I defended it before Eilenberg, Kadison, Morgenbesser and others in HamiltonHall in May 1963.

You studied in Columbia from February 1960 to June 1961, returning there for thePh.D. defense in May 1963. In the interim you went to Berkeley and Los Angeles.Why ?

Even though I had had an excellent course in mathematical logic from ElliottMendelson at Columbia, I felt a strong need to learn more set theory and logicfrom experts in that field, still of course with the aim of clarifying the foundationsof category theory and of physics. In order to support my family, and also becauseof my deep interest in mathematics teaching, I had taken up employment over thesummers of 1960 and 1961 with TEMAC, a branch of the Encyclopedia Britannica,which was engaged in producing high school text books in modern mathematicsin a new step-wise interactive format. In 1961, TEMAC built a new building nearthe Stanford University campus devoted to that project. Thus the further movewas not due to having lost a grant, but rather for those two purposes: in the Bayarea I could reside in Berkeley, follow courses by Tarski, Feferman, Scott, Vaught,and other leading set theorists, and also commute to Palo Alto to process the textbook which I was writing mainly at home.

Nor was my first destination in California the think tank referred to in Mac Lane’sbook. Rather, since my slow progress in writing my second programmed textbookwas not up to the speed which I thought TEMAC expected, I resigned from thatjob.

6Idem, ibidem, p. 351.

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A friend from the Indiana daysnow worked for the think tanknear Los Angeles, and was ableto persuade them to give me ajob. At the beginning I under-stood that the job would involvedesign of computer systems forverifying possible arms controlagreements; but when I finally gotthe necessary secret clearance,I discovered that other mat-ters were involved, related withthe Vietnam war. Mac Lane’saccount is essentially correctconcerning the way in which myfriend and fellow mathematicianBishop Spangler in the think tankbecame my supervisor and thengave me the opportunity to finishmy thesis on categorical univer-sal algebra. In February 1963,wanting very much to get out ofmy Los Angeles job to take up ateaching position at Reed College,

Dana Scott and F. William Lawvere (CT99)

I asked Eilenberg for a letter of recommendation. His very brief reply was that therequest from Reed would go into his waste basket unless my series of abstracts beterminated post haste and replaced by an actual thesis. This tough love had thedesired effect within a few weeks. Having defended the Ph.D. in May 1963, I wasable to leave the think tank and re-enter normal life as an assistant professor atReed College for the academic year 1963-64. En route to Portland I attended the1963 Model Theory meeting in Berkeley, where besides presenting my functorialdevelopment of general algebra, I announced that quantifiers are characterized asadjoints to substitution.

So, you spent the academic year 1963-64 as an assistant professor at Reed College.

At Reed I was instructed that the first year of calculus should concentrate onfoundations, formulas there being taught in the second year. Therefore, in spite ofalready having decided that the category of categories is the appropriate framework

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for mathematics in general, I spent several preparatory weeks trying to devise acalculus course based on Zermelo-Fraenkel set theory. However, a sober assessmentshowed that there are far too many layers of definitions, concealing differentiationand integration from the cumulative hierarchy, to be able to get through thoselayers in a year. The category structure of Cantor’s structureless sets seemed bothsimpler and closer. Thus, the Elementary Theory of the Category of Sets arosefrom a purely practical educational need, in a sort of experience that Saunders alsonoted: the need to explain daily for students is often the source of new mathemat-ical discoveries.

A theory of a category of Cantorian abstract sets has the same proof-theoreticstrength as the theory of a Category of Categories that I had initiated in the Intro-duction to my thesis. More objectively, sets can be defined as discrete categoriesand conversely categories can be defined as suitable finite diagrams of discrete sets,and the relative strengths thus compared. The category of categories is to be pre-ferred for the practical reason that all mathematical structures can be constructedas functors and in the proper setting there is no need to verify in every instancethat one has a functor or natural transformation.

After Reed I spent the summer of 1964 in Chicago, where I reasoned thatGrothendieck’s theory of Abelian categories should have a non-linear analoguewhose examples would include categories of sheaves of sets; I wrote down some ofthe properties that such categories should have and noted that, on the basis of mywork on the category of sets, such a theory would have a greater autonomy thanthe Abelian one could have (it was only in the summer of 1965 on the beach of LaJolla that I learned from Verdier that he, Grothendieck and Giraud had developeda full-blown theory of such “toposes”, but without the autonomy). Later, at theETH in Zurich ...

... where you stayed from September 1964 through December 1966 as visiting re-search scientist at Beno Eckmann’s Forschungsinstitut fur Mathematik ...

... there I was able to further simplify the list of axioms for the category of setsin a paper that Mac Lane then communicated to the Proceedings of the NationalAcademy of Sciences USA. There I also wrote up for publication the talk on “thecategory of categories as a foundation for mathematics” which I gave at the firstinternational meeting on category theory at La Jolla, California, 1965.

Which were the purposes of your elementary theory of the category of sets ?

It was intended to accomplish two purposes. First, the theory characterizes thecategory of sets and mappings as an abstract category in the sense that any model

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for the axioms that satisfies the additional non-elementary axiom of completeness,in the usual sense of category theory, can be proved to be equivalent to the categoryof sets. Second, the theory provides a foundation for mathematics that is quitedifferent from the usual set theories in the sense that much of number theory,elementary analysis, and algebra can apparently be developed within it even thoughno relation with the usual properties of ∈ can be defined.

Philosophically, it may be said that these developments supported the thesis thateven in set theory and elementary mathematics it was also true as has long beenfelt in advanced algebra and topology, namely that the substance of mathematicsresides not in Substance, as it is made to seem when ∈ is the irreducible predicate,but in Form, as is clear when the guiding notion is isomorphism-invariant struc-ture, as defined, for example, by universal mapping properties. As in algebra andtopology, here again the concrete technical machinery for the precise expressionand efficient handling of these ideas is provided by the Eilenberg-Mac Lane theoryof categories, functors and natural transformations.

A. Kock and F. W. Lawvere in Cafe Odeon, Zurich

(Fall of 1966; photo courtesy of A. Kock)

Let us return to Zurich.

At Zurich I had many discussions with Jon Beck and we collaborated on doctrines.The word “doctrine” itself is entirely due to him and signifies something which islike a theory, except appropriate to be interpreted in the category of categories,rather than, for example, in the category of sets. The “algebras” for a doctrinedeserve to be called “theories” because dualizing into a fixed algebra defines a

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semantics functor relating abstract generals and corresponding concrete generals.Jon was insistent on mathematical clarity and did much to encourage precisionin discussions and in the formulation of mathematical results. He noted that mystructure functor adjoint to semantics is analogous to Grothendieck’s cocycle defi-nition of descent in that both partially express the structure that inevitably ariseswhen objects are constructed by a functorial process, and which if hypothesizedhelps to reverse the process and discern the origin. Implementing this generalphilosophical notion of descent requires the choice of an appropriate “doctrine” oftheories in which the induced structure can be expressed.

Also from Zurich I attended a seminar in Oberwolfach where I met Peter Gabrieland learned from him many aspects not widely known even now of the Grothendieckapproach to geometry. In general the working atmosphere at the Forschungsinstitutwas so agreeable, that I later returned during the academic year 1968/69.

As an assistant professor in Chicago, in 1967, you taught with Mac Lane a courseon Mechanics, where “you started to think about the justification of older intu-itive methods in geometry”7. You called it “synthetic differential geometry”. Howdid you arrive at the program of Categorical Dynamics and Synthetic DifferentialGeometry ?

From January 1967 to August 1967 I was Assistant Professor at the University ofChicago. Mac Lane and I soon organized to teach a joint course based on Mackey’sbook “Mathematical Foundations of Quantum Mechanics”.

So, Mackey, a functional analyst from Harvard mainly concerned with the relation-ship between quantum mechanics and representation theory, had some relation tocategory theory.

His relation to category theory goes back much further than that, as Saundersand Sammy had explained to me. Mackey’s Ph.D. thesis displayed remarkablethinking of a categorical nature, even before categories had been defined. Specif-ically, the fact that the category of Banach spaces and continuous linear maps isfully embedded into a category of pairings of abstract vector spaces, together withthe definition and use of “Mackey convergence” of a sequence in a “bornological”vector space were discovered there and have played a basic role in some form innearly every book on functional analysis since. What is perhaps unfortunately notclarified in nearly every book on functional analysis, is that these concepts areintensively categorical in character and that further enlightenment would result ifthey were so clarified.

7Saunders Mac Lane, A Mathematical Autobiography, A K Peters, 2005.

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And the referee who, despite initial skepticism, permitted the first paper giving anexposition of the theory of categories to see the light of day in the TAMS in 1945,was none other than George Whitelaw Mackey.

Back to the origins of Synthetic Differential Geometry, where did the idea of orga-nizing such a joint course on Mechanics originate ?

Apparently, Chandra had suggested that Saunders give some courses relevant tophysics, and our joint course was the first of a sequence. Eventually Mac Lanegave a talk about the Hamilton-Jacobi equation at the Naval Academy in summer1970 that was published in the American Mathematical Monthly.

In my separate advanced lecture series, which was attended by my then studentAnders Kock, as well as by Mac Lane, Jean Benabou, Eduardo Dubuc, RobertKnighten, and Ulrich Seip, I began to apply the Grothendieck topos theory thatI had learned from Gabriel to the problem of simplified foundations of continuummechanics as it had been inspired by Truesdell’s teachings, Noll’s axiomatizations,and by my 1958 efforts to render categorical the subject of topological dynamics.

Beyond what I had learned from Gabriel at Oberwolfach on algebraic geometry asa gros topos, my particular contribution was to elevate certain ingredients, such asthe representing object for the tangent bundle functor, to the level of axioms so asto permit development unencumbered by particular construction. That particularingredient had apparently never been previously noted in the C-infinity category.It was immediately clear that the program would require development, in a similaraxiomatic spirit, of the topos theory of which I had heard in 1965 from Verdier onthe beach at La Jolla. Indeed, my appointment at Chicago had been encouragedalso by Marshall Stone who was enthusiastic about my 1966 observation that thetopos theory would make mathematical both the Boolean-valued models in gen-eral and the independence of the continuum hypothesis in particular. That theseapparently totally different toposes, involving infinitesimal motion and advancedlogic, could be part of the same simple axiomatic theory, was a promise in my 1967Chicago course. It only became reality after my second stay at the Forschungsin-stitut in Zurich, Switzerland 1968-69 during which I discovered the nature of thepower set functor in toposes as a result of investigating the problem of express-ing in elementary terms the operation of forming the associated sheaf, and after1969-1970 at Dalhousie University in Halifax, Nova Scotia, Canada, through mycollaboration with Myles Tierney.

You went to Dalhousie in 1969 with one of the first Killam professorships.

Indeed, and was able to have a dozen collaborators at my discretion, also supportedby Killam.

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And then you arrived, together with the algebraic topologist Myles Tierney, to theconcept of elementary topos. Could you describe us that collaboration with MylesTierney ?

Myles presented a weekly sem-inar in which the current stageof the work was described andindeed some of the work wasin the form of discussions inthe seminar itself: remarks bystudents like Michel Thiebaudand Radu Diaconescu weresometimes key steps. AlthoughI had been able to convincemyself in Zurich, Rome, andOberwolfach, that a finiteaxiomatization was possible, itrequired several steps of suc-cessive simplification to arrive

Myles Tierney and Dana Scott

(1971 conf. at Dalhousie, photo courtesy of R. Pare)

at the few axioms known now. The criterion of sufficiency was that by extendingany given category satisfying the axioms, it should be possible to build others bypresheaf and sheaf methods. The “fundamental theorem” of slices, followed byour discovery that left exact comonads also yield toposes, more than covered thepresheaf aspect. The concept of sheaves led to the conjecture that subtoposes wouldbe precisely parametrized by certain endomaps of the subobject classifier, and thiswas verified; those endomaps are now known as Lawvere-Tierney modal operators,and correspond classically to Grothendieck topologies. That the correspondingsubcategory of sheaves can be described in finite terms is a key technical feature,which was achieved by making explicit the partial-map classifier. That the theoryis elementary means that it has countable models and other features making itapplicable to independence results in set theory and to higher recursion, etc, buton the other hand Grothendieck’s theory of U -toposes is precisely included throughhis own technique of relativization together with additional axioms, such as thesplitting of epimorphisms and 2-valuedness, on U itself.

(By the way, those two additional axioms are positive – or geometrical– so thatthere is a classifying topos for models of them, a fact still awaiting exploitation byset theory.)

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Fred Linton and F. William Lawvere

(photo courtesy of R. Pare)

In 1971, official date of the birth of topos theory, unfortunately the dream team atDalhousie was dispersed. What happened, that made you go to Denmark ?

Some members of the team, including myself, became active against the Vietnamwar and later against the War Measures Act proclaimed by Trudeau. That Act,similar in many ways to the Patriot Act 35 years later in the US, suspended civilliberties under the pretext of a terrorist danger. (The alleged danger at the timewas a Quebec group later revealed to be infiltrated by the RCMP, the Canadiansecret police.) Twelve communist bookstores in Quebec (unrelated to the terror-ists) were burned down by police; several political activists from various groupsacross Canada were incarcerated in mental hospitals, etc. etc. I publicly opposedthe consolidation of this fascist law, both in the university senate and in publicdemonstrations. The administration of the university declared me guilty of “dis-ruption of academic activities”. Rumors began to be circulated, for example, thatmy categorical arrow diagrams were actually plans for attacking the administrationbuilding. My contract was not renewed.

And after a short period in Aarhus, you went to Italy. Why ?

Conditions in the Matematisk Institut were very agreeable, and the collaborationwith Anders Kock was very fruitful and enjoyable. However when the long northernnight set in, it turned out to be bad for my health, so I accepted an invitation fromPerugia. I still enjoy visiting Denmark in the summer.

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After a few years in Europe, you returned to the United States, for SUNY atBuffalo ...

John Isbell and Jack Duskin were able to persuade the dean that (contrary to themessage sent out by one of the Dalhousie deans) I was not a danger and mighteven be an asset.

In spite of your return to the USA, you kept close ties with the Italian mathematicalcommunity. In November 2003 there was a conference in Firenze (“Ramificationsof Category Theory”) to celebrate the 40th anniversary of your Ph.D. thesis8. Couldyou summarize the main ideas contained in it ?

Details are given in my commentary to the TAC Reprint (these Reprints are an ex-cellent source of other early material on categories). The main point was to presenta categorical treatment of the relation between algebraic theories and classes of al-gebras, incorporating the previous “universal” algebra of Birkhoff and Tarski in away applicable to specific cases of mathematical interest such as treated in books ofChevalley and of Cartan-Eilenberg. The presentation-free redefinition of both thetheories and the classes required explicit attention to the category of categories.

Ramifications of Category Theory, 2003

(photo by Andrej Bauer, used with permission)

8Functorial Semantics of Algebraic Theories, Reprinted in Repr. Theory Appl. Categ. 5 (2004) 1-121(electronic).

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In the Firenze conference there were talks both on mathematics and philosophy.You keep interested in the philosophy of mathematics ...

Yes. Since the most fundamental social purpose of philosophy is to guide educationand since mathematics is one of the pillars of education, accordingly philosophersoften speculate about mathematics. But a less speculative philosophy based onthe actual practice of mathematical theorizing should ultimately become one ofthe important guides to mathematics education.

As Mac Lane wrote in his Autobiography, “The most radical aspect is Lawvere’s no-tion of using axioms for the category of sets as a foundation of mathematics. Thisattractive and apposite idea has, as of yet, found little reflection in the communityof specialists in mathematical logic, who generally tend to assume that everythingstarted and still starts with sets”. Do you have any explanation for that attitude ?

The past 100 years’ tradition of “foundations as justification” has not helped math-ematics very much. In my own education I was fortunate to have two teachers whoused the term “foundations” in a common-sense way (rather than in the specula-tive way of the Bolzano-Frege-Peano-Russell tradition). This way is exemplifiedby their work in Foundations of Algebraic Topology, published in 1952 by Eilen-berg (with Steenrod), and the Mechanical Foundations of Elasticity and FluidMechanics, published in the same year by Truesdell. Whenever I used the word“foundation” in my writings over the past forty years, I have explicitly rejectedthat reactionary use of the term and instead used the definition implicit in thework of Truesdell and Eilenberg. The orientation of these works seemed to be“concentrate the essence of practice and in turn use the result to guide practice”.Namely, an important component of mathematical practice is the careful studyof historical and contemporary analysis, geometry, etc. to extract the essentialrecurring concepts and constructions; making those concepts and constructions(such as homomorphism, functional, adjoint functor, etc.) explicit provides power-ful guidance for further unified development of all mathematical subjects, old andnew.

Could you expand a little bit on that ?

What is the primary tool for such summing up of the essence of ongoing mathemat-ics? Algebra! Nodal points in the progress of this kind of research occur when, asin the case with the finite number of axioms for the metacategory of categories, allthat we know so far can be expressed in a single sort of algebra. I am proud to haveparticipated with Eilenberg, Mac Lane, Freyd, and many others, in bringing aboutthe contemporary awareness of Algebra as Category Theory. Had it not been for

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the century of excessive attention given to alleged possibility that mathematics isinconsistent, with the accompanying degradation of the F-word, we would still beusing it in the sense known to the general public: the search for what is “basic”.We, who supposedly know the explicit algebra of homomorphisms, functionals,etc., are long remiss in our duty to find ways to teach those concepts also in highschool calculus.

Having recognized already in the 1960s that there is no such thing as a heaven-givenplatonic “justification” for mathematics, I tried to give the word “Foundations”more progressive meanings in the spirit of Eilenberg and Truesdell. That is, I havetried to apply the living axiomatic method to making explicit the essential featuresof a science as it is developing in order to help provide a guide to the use, learning,and more conscious development of the science. A “pure” foundation which forgetsthis purpose and pursues a speculative “foundation” for its own sake is clearly aNON-foundation.

Foundations are derived from applications by unification and concentration, inother words, by the axiomatic method. Applications are guided by foundationswhich have been learned through education.

You are saying that there is a dialectical relation between foundations and applica-tions.

Yes. Any set theory worthy of the name permits a definition of mapping, domain,codomain, and composition; it was in terms of those notions that Dedekind andlater mathematicians expressed structures of interest. Thus, any model of such atheory gives rise to a category and whatever complicated additional features mayhave been contemplated by the theory, not only common mathematical proper-ties, but also most interesting “set theoretical” properties, such as the generalizedcontinuum hypothesis, Dedekind finiteness, the existence of inaccessible or Ulamcardinals, etc. depend only on this mere category.

During the past forty years we have become accustomed to the fact that founda-tions are relative, not absolute. I believe that even greater clarifications of foun-dations will be achieved by consciously applying a concentration of applicationsfrom geometry and analysis, that is, by pursuing the dialectical relation betweenfoundations and applications.

More recently, you have given algebraic formulations of such distinctions as ‘unityvs. identity’ of opposites, ‘extensive vs. intensive’ variable quantities, ‘spatial vs.quantitive’ categories ...

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Yes, showing that through the use of mathematical category theory, such questionslead not to fuzzy speculation, but to concrete mathematical conjectures and results.

It has been one of the characteristics of your work to dig down beneath the foun-dations of a concept in order to simplify its understanding. Here you are trulya descendant of Samuel Eilenberg, in his “insistence on getting to the bottom ofthings”. We vividly remember a lecture you presented in Coimbra to our under-graduate students. You have recently published a couple of textbooks9. Why do youfind it important enough to dedicate a significant amount of your time and effortto it ?

Many of my research publications are the result of long study of the two problems:(1) How to effectively teach calculus to freshmen. (2) How to learn, develop,and use physical assumptions in continuum thermomechanics in a way which isrigorous, yet simple.

F. William Lawvere and Stephen Schanuel

(Sydney, 1988; photo courtesy of R. Walters)

In other words, the results themselves can only be building blocks in an answer tothe question: “How can we take concrete, pedagogical steps to narrow the enor-mous gap in 20th century society between the fact that: (a) everybody must usetechnology which rests on science, which in turn depends on mathematics; yet (b)

9F. W. Lawvere and R. Rosebrugh, Sets for Mathematics, Cambridge University Press, Cambridge, 2003;F. W. Lawvere and S. Schanuel, Conceptual Mathematics. A First Introduction to Categories, CambridgeUniversity Press, Cambridge, 1997.

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only a few have a working acquaintance with basic concepts of modern mathemat-ics such as retractions, fixed-point theorems, morphisms of directed graphs and ofdynamical systems, Galilean products, functionals, etc.”

Only armed with such concepts can one hope to respond with confidence to themyriad of methods, results, and claims which in the modern world are associatedwith mathematics. With Stephen Schanuel I have begun to take up the challengeof that question in our book Conceptual Mathematics which reflects the ongoingwork of many mathematicians.

What is your opinion on the Wikipedia article about you ?

The disinformation in the original version has been largely removed, but muchremains in other articles about category theory.

We have recently celebrated Kurt Godel’s 100th birthday. What do you think aboutthe extra-mathematical publicity around his incompleteness theorem ?

In Diagonal arguments and Cartesian closed categories10 we demystified the incom-pleteness theorem of Godel and the truth-definition theory of Tarski by showingthat both are consequences of some very simple algebra in the Cartesian-closedsetting. It was always hard for many to comprehend how Cantor’s mathematicaltheorem could be re-christened as a “paradox” by Russell and how Godel’s theoremcould be so often declared to be the most significant result of the 20th century.There was always the suspicion among scientists that such extra-mathematicalpublicity movements concealed an agenda for re-establishing belief as a substitutefor science. Now, one hundred years after Godel’s birth, the organized attemptsto harness his great mathematical work to such an agenda have become explicit11.

You have always been concerned in explaining how to describe relevant mathemati-cal settings and facts in a categorical fashion. Is category theory only a language ?

No, it is more than a language. It concentrates the essential features of centuriesof mathematical experience and thus acts as indispensible guide to further devel-opment.

What have been for you the major contributions of category theory to mathematics ?

10Reprinted in Repr. Theory Appl. Categ. 15 (2006) 1-13 (electronic).11The controversial John Templeton Foundation, which attempts to inject religion and pseudo-science

into scientific practice, was the sponsor of the international conference organized by the Kurt GodelSociety in honour of the celebration of Godel’s 100th birthday. This foundation is also sponsoring aresearch fellowship programme organized by the Kurt Godel Society.

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First, the work of Grothendieck in his Tohoku’s paper12. Nuclear spaces was oneof the great inventions of Grothendieck. By the way, Silva worked a lot on thesespaces and Grothendieck’s 1953 paper on holomorphic functions13 was inspired bya 1950 paper of Silva14.

The concept of adjoint functors, discovered by Kan in the mid 1950’s, was also amilestone, rapidly incorporated as a key element in Grothendieck’s foundation ofalgebraic geometry and in the new categorical foundation of logic and set theory.

I may also mention Cartesian closedness, the axiomatization of the category ofcategories, topos theory ... Cartesian closed categories appeared the first time inmy Ph.D. thesis, without using the name. The name appeared first in Kelly andEilenberg’s paper15. I don’t exactly agree with the word “Cartesian”. Galileo isthe right source, not Descartes.

You are regarded by many people as one of the greatest visionaries of mathematicsin the beginning of the twentieth first century. What are your thoughts on the futuredevelopment of category theory inside mathematics ?

I think that category theory has a role to play in the pursuit of mathematicalknowledge. It is important to point out that category theorists are still findingstriking new results in spite of all the pessimistic things we heard, even 40 yearsago, that there was no future in abstract generalities. We continue to be surprisedto find striking new and powerful general results as well as to find very interestingparticular examples.

We have had to fight against the myth of the mainstream which says, for example,that there are cycles during which at one time everybody is working on generalconcepts, and at another time anybody of consequence is doing only particularexamples, whereas in fact serious mathematicians have always been doing both.

One should not get drunk on the idea that everything is general. Category theoristsshould get back to the original goal: applying general results to particularities andto making connections between different areas of mathematics.

12A. Grothendieck, Sur quelques points d’algebre homologique, Tohoku Math. J. 9 (1957) 119-121.13A. Grothendieck, Sur certains espaces de fonctions holomorphes, I, J. Reine Angew. Math. 192 (1953)

35-64.14J. Sebastiao e Silva, Analytic functions and functional analysis, Portugaliae Math. 9 (1950) 1-130.15S. Eilenberg and G. M. Kelly, Closed categories, in: Proc. Conf. Categorical Algebra (La Jolla, Calif.,

1965), pp. 421-562, Springer, 1966.

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F. William Lawvere and Maria Manuel Clementino

(Braga, March 2007)

Francis William Lawvere (born February 9, 1937 in Muncie, Indiana) is a mathe-matician well-known for his work in category theory, topos theory, logic, physicsand the philosophy of mathematics. He has written more than 60 papers in thesubjects of algebraic theories and algebraic categories, topos theory, logic, physics,philosophy, computer science, didactics, history and anthropology, and has threebooks published (one of them with translations into Italian and Spanish), withthree more in preparation at this moment. He also edited three volumes of theSpringer series Lecture Notes in Mathematics and supervised twelve Ph.D. theses.The electronic series Reprints in Theory and Applications of Categories includesreprints of seven of his fundamental articles, with author commentaries, amongthem his Ph.D. dissertation and his full treatment of the category of sets.

At the 1970 International Congress of Mathematicians in Nice he introduced an al-gebraic version of topos theory which unified geometry and set theory. Worked outin collaboration with Myles Tierney, this theory has since been developed furtherby many people, with applications to several fields of mathematics. Two of thosefields had previously been introduced by Lawvere: (1) His 1967 Chicago lectures

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(published 1978) on categorical dynamics had shown how toposes with specifiedinfinitesimal objects can provide a flexible geometric background for models ofcontinuum physics, which led to a new subject known as Synthetic DifferentialGeometry; (2) In his 1967 Los Angeles lecture, and his 1968 papers on hyperdoc-trines and adjointness in foundations, Lawvere had launched and developed thefield of categorical logic, which has since been widely applied to geometry andcomputer science. Those ideas were indispensable for his 1983 simplified proof ofthe existence of entropy in non-equilibrium thermomechanics.

Many of Lawvere’s research publications result from efforts to improve the teachingof calculus and of engineering thermomechanics. In particular, it was his 1963 ReedCollege course in the foundations of calculus which led to his 1964 axiomatizationof the category of sets and ultimately to the elementary theory of toposes.

Professor Lawvere studied with Clifford Truesdell and Max Zorn at Indiana Uni-versity and completed his Ph.D. at Columbia in 1963 under the supervision ofSamuel Eilenberg. Before completing his Ph.D., Lawvere spent a year in Berkeleyas an informal student of model theory and set theory, following lectures by AlfredTarski and Dana Scott. During 1964-1966 he was a visiting research professor atthe Forschungsinstitut fur Mathematik at the ETH in Zurich. He then taught atthe University of Chicago, working with Mac Lane, and at the City University ofNew York Graduate Center (CUNY), working with Alex Heller. Back in Zurich for1968-69 he proposed elementary (first-order) axioms for toposes generalizing theconcept of the Grothendieck topos. Dalhousie University in 1969 set up a group ofKillam-supported researchers with Lawvere at the head; but in 1971 it terminatedthe group because of Lawvere’s political opinions (namely his opposition to the1970 use of the War Measures Act).

Then Lawvere went to the Institut for Matematiske in Aarhus (1971-72) and ran aseminar in Perugia, Italy (1972-1974) where he especially worked on various kindsof enriched category. From 1974 until his retirement in 2000 he was professorof mathematics at the University at Buffalo, often collaborating with StephenSchanuel. There he held a Martin professorship (1977-82). He was also a visitingresearch professor at the IHES Paris (1980-81).

He is now Professor Emeritus of Mathematics and Adjunct Professor Emeritus ofPhilosophy at the State University of New York at Buffalo and continues to workon his 50-year quest for a rigorous and flexible framework for the physical ideas ofTruesdell and Walter Noll, based on category theory.

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His personal view of mathematicsand physics, based on a broad anddeep knowledge, keeps influencingmathematicians and attractingexperts from other areas to Math-ematics. This influence was veryapparent in the honouring sessionthat took place in the last Interna-tional Category Theory Conference(Carvoeiro, Portugal, June 2007),on the occasion of his 70th Birthday,through spontaneous and intensetestimonies of both senior mathe-maticians and young researchers.Indeed, besides his extraordinaryqualities as a mathematician, wewish to stress the care and effortshe puts into the guidance of stu-dents and young researchers, which

Celebration of the 70th birthday of F. W. Lawvere

(CT07, Carvoeiro, Algarve)

we could confirm in Coimbra when he gave a lecture on Category Theory to un-dergraduate students, and again in the dialog we were very honoured to be partof, during the preparation of this interview.

Lecturing in Coimbra, March 1997

(photo by Manuela Sobral)

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